It was about the Russians, French and British fighting over control or controlling influence over the declining Ottoman Empire. Britain wanted to maintain it's naval and economic supremacy in the Mediterranean, as well as retain that presence in the Black Sea as well. The Crimean War wasn't about the British wanting to spite the Russians, there was something in particular in the region at stake.
Alaska, on the otherhand, was too big to sufficiently defend, and too big to want to capture all at once, and seemingly of little value at the time. This was long before they found oil there. Before, that Alaska was known for it's fur-trapping.
Besides, Britain already had Canada. Plenty of fur-trapping to be had.